Regeneration and Self-Maintaining of Coherent Structures - Transport and Coherent Structures in Wall Turbulence - page 367

by deriving and integrating the above equation on a
piecewise basis:

∂
−∧ ∂

ω

( xx ') ( xt ', )

∂ u 1 x '

∫

(,) x t =−

i

d V ( ) x [5.105]

3

∂ x 4 π

xx − '

i

R

where x i corresponds to x , y and z, respectively, for

and 3. From this, we deduce: i =

1, 2

∂ ω

∂ ω

y

( xx − ') − ( yy − ')

x

∂ w 1 ∂ x ' ∂ x '

∫

(,) x t =− d x dy dz ' ' '

∂ x 4 π

3/2

⎡

( ) (

2

) (

2

)

2

⎤

xx −+−+− ' yy ' zz '

R

⎣ ⎦

[5.106]

We can see that wx - which, remember, plays a part in

∂∂

the generation of in terms of predominant production -

can be engendered by

ω

x

. The process
under discussion in this study is based on the first scenario.
Indeed, the streaks play a fundamental role both in bypass
transition and in the physics of near-wall turbulence, and
the layers

, and/or by ω∂∂ x

ω∂∂ x

y x

delimiting the streaks constitute a key element
in the regeneration of the structures. Let us look again at
Figure 5.44 and suppose that there is a variation in direction
x of the layers

ω

y

associated with the “mother”
vortex A , situated, respectively, at

and

ω + ω −

yA yA

, and
which are, for the time being, symmetrically distributed
around

and

z < 0 z > 0

experience an absolutely identical variation in x , no zone

. We can easily deduce that if and

ω + ω −

z = 0

yA yA